Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions ₊(q) and ₋(q) of (q) to the set ₊(q) of secant (hyperbolic) lines and to the set ₋(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme ₋(q) is pseudocyclic.We further show that the coherent configurations (q ²) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme ₊(q ²), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes ₊(q ²) and ₋(q ²). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.